We study two-magnon spectra in resonant inelastic x-ray scattering (RIXS) from the Heisenberg antiferromanets by extending the formula of Nomura and Igarashi (Phys. Rev. B 71, 035110 (2005)). The core-hole potential in the intermediate state of RIXS gives rise to a change in the exchange coupling between 3d electrons, leading to an effective interaction between the core hole and spins of 3d electrons. We derive a formula suitable to calculate the two-magnon RIXS intensities, replacing the bare core-hole potential responsible to charge excitations by this effective interaction creating two magnons in our previous formula. It consists of two factors, one of which determines the incident-photon-energy dependence and another is the two-magnon correlation function. We evaluate the former factor for La 2 CuO 4 in terms of the density of states of the 4p states obtained by the band calculation. We also calculate the two-magnon correlation function as a function of energy loss ω and momentum transfer q of the Heisenberg model on a square lattice, by summing up the ladder diagrams after transforming the magnon-magnon interaction into a separable form.The calculated spectra form a broad peak around ω = 3J for S = 1/2 on the magnetic Brillouin zone boundary and vanish at q = (0, 0) and (π, π). Such momentum dependence of the RIXS spectra could provide an excellent opportunity to study the dynamics in the Heisenberg model.

We study the effects of quantum fluctuations on excitation spectra in the two-dimensional Heisenberg antiferromagnet by means of the 1/S expansion. We calculate the spin-wave dispersion and the transverse dynamical structure factor up to the second order of 1/S in comparison with inelastic neutron scattering experiments. The spin-wave energy at momentum (π, 0) is found to be about 2% smaller than that at (π/2, π/2) due to the second-order correction. In addition, we study the dimensional crossover from two dimensions to one dimension by weakening exchange couplings in one direction. It is found that the second-order correction becomes large with approaching the quasi-one dimensional situation and makes the spin-wave energy approach to the des CloizeauxPearson boundary for S = 1/2. The transverse dynamical structure factor is also calculated up to the second order of 1/S. It is shown that the intensity of spin-wave peak is strongly reduced while the intensity of three-spin-wave continuum becomes large and exceeds that of the spin-wave peak in the quasi-one dimensional situation. The physics of low-dimensional spin systems has attracted much interest for past decades.Although the quantum fluctuation is expected to be large in two dimensions, there exist strong evidences that the quantum Heisenberg antiferromagnet (QHAF) with nearestneighbor coupling in a square lattice exhibits the Néel long-range order at zero temperature. 1Under the presence of the long-range order, the linear spin-wave (LSW) theory works rather well.2,3 A natural way to include quantum fluctuations is an expansion in terms of 1/S, where S is the magnitude of spin, because the LSW theory is made up of a leading-order in the 1/S expansion. Such attempts have been done and turned out to be useful. 4,5,6,7,8,9 In our previous paper, 5 basing on the Holstein-Primakoff transformation, 10 we calculated corrections up to the second order of 1/S in various physical quantities such as the spin-wave dispersion, the sublattice magnetization, the perpendicular susceptibility, and the spin-stiffness constant. We also calculated the dynamical structure factors of the transverse and the longitudinal components up to the second order of 1/S. 6 At that time, however, available experimental data of inelastic neutron scattering (INS) were limited to the momentum transfer in a narrow region of the Brillouin zone (BZ). Therefore, our study was just a demonstration of usefulness of the 1/S expansion. Now that INS experiments provide us the information of the excitation spectra in the whole BZ, it may be interesting to calculate the excitation spectra in the whole BZ in comparison with recent experiments. We show that the second-order correction makes the spin wave energy at momentum (π, 0) about 2%smaller than that at (π/2, π/2). This difference is smaller than the value 7 ∼ 9% obtained by the series expansion 11,12 and the Monte Carlo simulation. 13 We have so far not been able to find why the 1/S expansion within the second order gives different results, si...

We study the magnetic excitation spectra in L-edge resonant inelastic x-ray scattering (RIXS) from undoped cuprates. We analyze the second-order dipole-allowed process that the strong perturbation works through the intermediate state in which the spin degree of freedom is lost at the core-hole site. Within the approximation neglecting the perturbation on the neighboring sites, we derive the spin-flip final state in the scattering channel with changing the polarization, which leads to the RIXS spectra expressed as the dynamical structure factor of the transverse spin components. We assume a spherical form of the spin-conserving final state in the channel without changing the polarization, which leads to the RIXS spectra expressed as the "exchange"-type multispin correlation function. Evaluating numerically the transition amplitudes to these final states on a finite-size cluster, we obtain a sizable amount of the transition amplitude to the spin-conserving final state in comparison with that to the spin-flip final state. We treat the itinerant magnetic excitations in the final state by means of the 1/S expansion method. Evaluating the higher-order correction with 1/S, we find that the peak arising from the one-magnon excitation is reduced with its weight, and the continuous spectra arising from the three-magnon excitations come out. The interaction between two magnons is treated by summing up the ladder diagrams. On the basis of these results, we analyze the L 3 -edge RIXS spectra in Sr 2 CuO 2 Cl 2 in comparison with the experiment. It is shown that the three-magnon excitations as well as the two-magnon excitations give rise to the intensity in the high-energy side of the one-magnon peak, making the spectral shape asymmetric with wide width, in good agreement with the experiment.